Discrete Counting Problems

[intervade]

Ok, I'm given this SSN problem. There are a number of different questions they ask and they are all pretty straight forward, for example, how many SS numbers are there.. (10^9). I'm struggling on some of the more difficult questions for exmple.. How many have exactly one 8 in the number? and How many read the same backward and foward(e.g. 122979221).

For the first one, would it be ok to assume and put the 8 in the first slot. There for, I have 9^9 * 1 ? And for the second one, Im completely lost.

 

[galactus]

How many SSN have exactly one 8 in the number?

There are 9 places to put the 8. (i.e. 128949776). Now, since it is placed, there are 9 choices for the digits to place in 8 other spots.

\(\displaystyle 9\cdot 9^{8}=9^{9}\)

How many SSN read the same backward and foward(e.g. 122979221).

Start from the left and go right. The first digit has 10 choices. That means the digit on the far right has to be the same.

The second digit, there are 10 choices. That means the next to last digit must be the same as that.

The third digit, there are 10 choices. That means the third from the end has to match it.

The fourth digit, there are 10 digits. The fourth from the end is the same.

But, the fifth digit, the one in the middle will have to be different than the other 8. So, there are 9 choices.

\(\displaystyle 9\cdot 10^{4}\)

 

[mmm4444bot]

I'm thinking that clarification is needed regarding what constitutes a "SSN" in this exercise.